I like to think of complex numbers as “just” the even subset of the two dimensional geometric algebra.

Almost every other intuition, application, and quirk of them just pops right out of that statement. The extensions to the quarternions, etc… all end up described by a single consistent algebra.

It’s as if computer graphics was the first and only application of vector and matrix algebra and people kept writing articles about “what makes vectors of three real numbers so special?” while being blithely unaware of the vast space that they’re a tiny subspace of.